Asked by Tyler
Which of the following is rational?(1 point) Responses 3–√ +3–√ 3–√ +3–√ 13√ the fraction with numerator 1 and denominator square root of 3 3 −−√ × 3–√ 3 −−√ × 3–√ 3√1
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Answered by
GPT-4o mini
To determine which of the given expressions is rational, let's analyze each one step-by-step.
1. **\( \sqrt{3} + \sqrt{3} \)**: This simplifies to \( 2\sqrt{3} \), which is irrational since \( \sqrt{3} \) is irrational.
2. **\( \frac{1}{\sqrt{3}} \)**: This is also irrational, but if we rationalize the denominator, we multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \), which gives us \( \frac{\sqrt{3}}{3} \). This expression is still irrational.
3. **\( \sqrt{3} \times \sqrt{3} \)**: This gives us \( 3 \), which is a rational number.
4. **\( \sqrt{3} \cdot 1 \)**: This is simply \( \sqrt{3} \), which is irrational.
Based on the analysis, the only rational expression is **\( \sqrt{3} \times \sqrt{3} = 3 \)**. So the correct answer is:
**3** (the product of \( \sqrt{3} \) with itself).
1. **\( \sqrt{3} + \sqrt{3} \)**: This simplifies to \( 2\sqrt{3} \), which is irrational since \( \sqrt{3} \) is irrational.
2. **\( \frac{1}{\sqrt{3}} \)**: This is also irrational, but if we rationalize the denominator, we multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \), which gives us \( \frac{\sqrt{3}}{3} \). This expression is still irrational.
3. **\( \sqrt{3} \times \sqrt{3} \)**: This gives us \( 3 \), which is a rational number.
4. **\( \sqrt{3} \cdot 1 \)**: This is simply \( \sqrt{3} \), which is irrational.
Based on the analysis, the only rational expression is **\( \sqrt{3} \times \sqrt{3} = 3 \)**. So the correct answer is:
**3** (the product of \( \sqrt{3} \) with itself).
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